1. Platos poor city and rich city Philippe Woloszyn Chargé de Recherche CNRS ESO UMR CNRS 6590, Université de Haute Bretagne Rennes2, Maison de la Recherche en Sciences Sociales, 35043 Rennes, France [email_address] 10th international Conference of Territorial Intelligence Liege 2011 From economical to social entropy measure. Towards an inequality multidimensional evaluation.

2. poor city An inequality measure has been inspired by Plato: The Plato inequality refers to Plato's "two cities", a city devided into at least two quantiles: 20% of the citizens control a share of 80% of all resources and rich city

3. …Pareto distribution Fractal distribution: segments of the curves are self-similar Pareto « 80-20 » law, according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is α = log 4 5. Probability density function for various α with x m = 1. Cumulative distribution functions for various α with x m = 1

4. Pareto-Zipf Law [Mandelbrot (1965)] As Pareto distributions are continuous probability distributions, Zipf law is the discrete counterpart of the Pareto distribution. Informational impedance adaptation level between economical and social dimensions

5. After Plato's two-quantile-societies and Pareto-Zipf law, Time series indicators has been constructed, such as Robin Hood Index (Hoover inequality) Gini inequality and Entropy measures Inequality Issuization Measure of redundancy, lack of diversity, isolation, segregation, inequality, non-randomness, and compressibility, describes the maximum possible entropy of the data minus the observed entropy, through generalized entropy index computation It measures an entropic "distance" between population and an hypothetic "ideal" egalitarian state. Derived from Shannon’s entropy mesurement, entropy is a measure of economical, social and environmental « order »

6. Entropy as a sustainability measure Econogical weight Socioecological weight Socioeconomical weight Economical Entropy Social Entropy Environmental Entropy Vulnerability Resilience DEVELOPMENT NEEDS RESOURCE

7. Applications of entropy statistics The parameter α in the GE class represents the weight given to distances between incomes at different parts of the income distribution Organisation theory Saviotti (1988) Technological evolution ( innovation studies Frenken et al. 1999) Regional industrial diversification (Hackbart and Anderson 1975; Attaran 1985) Corporate diversification and profitability (Jacquemin and Berry 1979; Palepu 1985; Hoskisson et al. 1993) Industrial concentration (Hildenbrand and Paschen 1964; Finkelstein and Friedberg 1967; Theil 1967: 290-291) GE(1) is Theil’s T index : Income inequality (Theil 1967: 91-134 and 1972: 99-109)

9. Source: Conceição, P. and Ferreira, P. (2000) : The Young Person’s Guide to the Theil Index: Suggesting Intuitive Interpretations and Exploring Analytical Applications, UTIP Working Paper N°14, 54p World Inequality: Population and Income Shares of the 54 Richest and 54 Poorest Countries in the World in 1970. Simulation of the Evolution of Inequality Measures as the Shares of Income Change. Theil Index Deconstruction

10. Time series spatialization: REGIONAL PATTERNS OF INEQUALITY Fractal behavior of the Theil Index (attractor ?) Vertical cut, with origin in individual i, intercepts the boundary of individual i (its source point), but also the boundaries of group l and group a.

11. Social Entropy Econogical weight Socioecological weight Socioeconomical weight Economical Entropy Environmental Entropy Vulnerability Resilience DEVELOPMENT NEEDS RESOURCE

12. Example classification using numerical techniques. The top row shows how the system is clustered at several levels, parameterized by taxonomic level h. The classification is summarized in a taxonomic tree, or dendrogram (bottom). Social entropy social entropy (computed using Shannon’s information entropy formulation (Shannon, 1949)) as an appropriate measure of diversity TUCKER BALCH: Hierarchic Social Entropy: An Information Theoretic Measure of Robot Group Diversity, Autonomous Robots 8, 209–237, 2000

13. branching structure of the dendrograms for these two societies is the same. However, the more compact distribution of elements in the system on the upper right is reflected in the branches being compressed towards the bottom of the corresponding dendrogram (lower right). Hierarchic Social Entropy Metric Because hierarchic entropy is scale invariant it can distinguish between the two societies regardless of the value of x.

14. Application: Inequality trends comparison among European countries KRZANOWSKI, W.J. (1979) “Between-groups comparison of principal components”. Journal of the American Statistical Association, 74, 703-707. Correction note (1981), 76, 1022. KRZANOWSKI, W.J. (1982) “Between-groups comparison of principal components – some sampling results”. Journal of Statistical Computation and Simulation, 15, 141-154. Groups of countries derived from the classification according to their inequality level Dendrogram of the countries’ common space based inequality index European Community Household Panel (ECHP)

15. From economics to social: the « social capital » ecosocial combination. (equitable/social justice) Relationship Between Bridging Social Capital and Governance Source: Narayan (1999) Ecosocial indexing

16. Econogical weight Socioecological weight Socioeconomical weight Economical Social Environmental Vulnerability Resilience DEVELOPMENT NEEDS RESOURCE Ecosocial entropy

17. The case α = ∞ corresponds to perfectly equal distribution (G = 0) and the α = 1 line corresponds to complete inequality (G = 1) Gini coefficient a measure of the deviation of the Lorenz curve from the equidistribution plot of the fraction of total income held by a given fraction of the population (Heaps et al., 1998, Aaberge 2005). Income patterns for Pareto distribution Lorenz Curve & Gini Coefficient Gini coefficient for a Pareto distribution Lorenz curves for Pareto distributions.

18. Foster, Greer, and Thorbecke (1984) class of poverty measures Pareto distribution of income patterns: a powerty measurement? Robin Hood Index, hunger line and Hoover inequality Measure of the maximum vertical distance between the Lorenz curve and the line of perfect equality (45° line) Three dimensions of poverty—headcount, shortfall, and inequality among the poor— α the parameter z the distance to the hunger Line i-> n the number of individuals

19. Hunger Line: the social risk perception Sources: Current hunger levels from FAO (1997a); Incomes from WRI (1996b), Heaps, et al. (1998) Hunger Lines vs. Mean Income Hunger line tend to rise as average incomes do (Ravallion et al . , 1991; World Bank, 1990)

20. From Plato inequality To redistributive aggression? Tresspassing the hunger Line

21. Conclusion: Inequality is leading to battlefields Unfair wealth distribution leads to social aggression Ecosocial entropy measure is a risk indicator

22. Pareto, Vilfredo, Cours d’Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino, Librairie Droz, Geneva, 1964, pages 299–345. Seal, H. (1980). Survival probabilities based on Pareto claim distributions. ASTIN Bulletin, 11, 61–71. M. E. J. Newman (2005). "Power laws, Pareto distributions and Zipf's law". Contemporary Physics 46 (5): 323–351 Aaberge, R. (2000): Characterizations of Lorenz curves and income distributions, Social Choice and Welfare 17, 639-653. Aaberge, R. (2001): Axiomatic characterization of the Gini coefficient and Lorenz curve orderings, Journal of Economic Theory, 101, 115-132. Reed, W.J., 2000 a. The Pareto law of incomes – an explanation and an extension. Submitted to Journal of Business and Economic Statistics. Reed, W.J., 2000 b. On the rank-size distribution for human settlements. Submitted to Regional Science and Urban Economics. Gabaix, X., 1999. Zipf’s law for cities: an explanation. Quarterly Journal of Economics, 114 739-767. Brakman, S, H.Garretsen, C. Van Marrewijk and M. van den Berg, 1999. The return of Zipf: Towards a further understanding of the rank-size distribution. Journal of Regional Science, 39:739-767.